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Low-Complexity Sparse FIR Channel Shortening
Ahmad Gomaa, Student Member, IEEE, and Naofal Al-Dhahir, Fellow, IEEE
The University of Texas at Dallas, Richardson, TX, USA
Abstract—The complexity of maximum-likelihood (ML) or
maximum-a-posteriori (MAP) detectors grows exponentially with
the number of channel impulse response (CIR) taps. This makes
the implementation of ML or MAP detectors over broadband
channels with long CIRs prohibitively complex. Channel short-
ening is a widely-used technique to solve this problem by
implementing a front-end ﬁnite impulse response (FIR) ﬁlter to
shorten the CIR. In this paper, we propose a novel approach
based on compressive sensing theory to design low-complexity
FIR channel shortening ﬁlters. The superiority of our new
approach is proven analytically and illustrated via simulations.
I. INTRODUCTION
Broadband communication is characterized by long channel
impulse responses (CIRs) which introduce signal process-
ing implementation challenges for both single-carrier (SC)
systems and multi-carrier modulation (MCM) systems. In
SC systems, the complexity of maximum-likelihood sequence
estimation (MLSE) grows exponentially with the number of
CIR taps. In MCM systems, a guard sequence is inserted
after each input block to prevent inter-block interference. The
guard sequence length must be greater than or equal to the
CIR length. For channels with severe inter-symbol interference
(ISI), the resulting guard sequence overhead causes signiﬁcant
data rate loss.
As a remedy for this problem, a front-end equalizer, com-
monly known as a channel shortening equalizer (CSE), is
designed such that the cascade of the long CIR and the
equalizer is equivalent to a short target impulse response
(TIR). Several TIR design criteria have been investigated in
the literature. In [1], the TIR was chosen to be a truncated
version of the original IR. A monocity condition on the TIR
was suggested in [2]. In [3], the mean-square error (MSE)
between the TIR and the cascade of the CIR and the CSE was
minimized subject to a unit-energy constraint (UEC) on the
TIR. Finite-length equalizers were proposed in [4] for MCM
systems under a unit-tap constraint (UTC). In [5], the UTC
and the UEC criteria were uniﬁed under a single framework
that lends itself to other constraints as well. In [6], the CSE
designs were generalized to multiple-input multiple-output
(MIMO) channels. In [7], channel shortening is performed
blindly without channel knowledge at the receiver.
In [5], the TIR taps were chosen to be contiguous and the
CSE is designed using the minimum MSE (MMSE) criterion.
In this paper, we relax this condition by allowing the TIR taps
and the CSE taps to be noncontiguous and sparse. We show
how the tap locations and weights can be computed using
compressive sensing (CS) theory [8], [9]. In addition, we prove
analytically that the resulting MMSE is reduced or at least
remains the same as in the contiguous case while reducing the
This work is supported by a gift from RIM Inc.
implementation complexity signiﬁcantly. A Viterbi algorithm
(VA) was proposed in [10] for MLSE of signals in sparse
channels with noncontiguous taps. A parallel-trellis implemen-
tation of the VA was presented in [11] for sparse channels
where the taps are noncontiguous but uniformly spaced. Turbo
equalization for channels with noncontiguous taps has been
proposed in [12], [13]. Furthermore, we use CS theory to
design a sparse shortening equalizer where the number of its
taps is reduced at a small performance loss. Reducing the
number of equalizer taps automatically reduces the number
of complex multiplications required to ﬁlter (equalize) the
received signal. The rest of this paper is organized as follows.
In Section II, we provide a review of CS theory and describe
the signal model. The sparse channel shortening problem is
formulated in Section III and the sparse design of the CSE
is described in Section IV. In Section V, we present the
simulation results. Finally, the conclusion is given in Section
VI. Notations: Unless otherwise stated, lower and upper case
bold letters denote vectors and matrices, respectively. The
matrix I denotes the identity matrix and its size is denoted by
the subscript. The matrix 0
m×n
denotes the all-zero matrix
of size m × n. Also, ( )
H
, ( )
∗
, and ( )
−1
denote the
matrix complex-conjugate transpose, the complex conjugate,
and the matrix inverse operations, respectively. The portion of
the vector x starting from the index a and ending at the index
b is denoted by x
a:b
.
II. CS BACKGROUND AND SIGNAL MODEL
A. Compressive Sensing Background
CS theory [8], [9] asserts that we can recover a sparse vector
x ∈ C
N
from a measurement vector y ∈ C
M
where M N.
In other words, the exact solution of the under-determined
system of equations y = Ax + z can be computed where
A denotes the M × N measurement matrix and z ∈ C
M
is
a zero-mean random noise vector. The word “sparse” means
that x contains few nonzero elements. The sparse vector x
is recovered by solving the following l
1
-norm constrained
minimization problem [14]
min
˜x∈C
N
˜x
1
subject to y −A˜x
2
≤ (1)
where .
1
and .
2
denote the l
1
-norm and the l
2
-norm,
respectively, and is chosen such that it bounds the amount
of noise in the measurements. In fact, the above program
searches for the sparsest vector ˜x such that y − A˜x
2
2
≤ .
Furthermore, the convex optimization problem in (1) is a
second-order cone program and can be solved efﬁciently [15].
2
x
k
h
k
n
k
W
e
k y
k
b
Fig. 1. System block diagram
B. Signal Model
We consider the case of a linear, time invariant, dispersive,
and noisy communication channel. The complex-valued equiv-
alent baseband signal model is used. Assuming an oversam-
pling factor of l, the received samples have the standard form
y
k
=
ν
¸
m=0
h
m
x
k−m
+ n
k
(2)
where h
m
is the CIR whose memory is denoted by ν and n
k
is the additive noise. The quantities {y, h
m
, n} are l × 1 col-
umn vectors corresponding to the l time (fractionally-spaced)
samples per symbol in the assumed temporally oversampled
channel model. Furthermore, x
k−m
is the transmitted symbol
at time (k −m).
Grouping {y
k
} over a block of N
f
symbol periods, the
input-output relation in (2) can be expressed in the following
matrix notation
⎡
⎢
⎢
⎢
⎣
y
k
y
k−1
.
.
.
y
k−N
f
+1
⎤
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎣
h
0
h
1
. . . h
ν
0 . . . 0
0 h
0
h
1
. . . h
ν
0 . . .
.
.
.
.
.
.
.
.
.
0 . . . 0 h
0
h
1
. . . h
ν
⎤
⎥
⎥
⎥
⎦
.
⎡
⎢
⎢
⎢
⎣
x
k
x
k−1
.
.
.
x
k−N
f
−ν+1
⎤
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎣
n
k
n
k−1
.
.
.
n
k−N
f
+1
⎤
⎥
⎥
⎥
⎦
(3)
or more compactly
y
k:k−N
f
+1
= Hx
k:k−N
f
−ν+1
+ n
k:k−N
f
+1
(4)
The (N
f
+ ν) ×(N
f
+ ν) input correlation matrix is deﬁned
by
R
xx
≡ E

.
Both R
xx
and R
nn
are assumed to be positive-deﬁnite (non-
singular) correlation matrices. Furthermore, the output-input
cross-correlation and the output auto-correlation are, respec-
tively, given by
R
yx
≡ E

y
k:k−N
f
+1
x
H
k:k−N
f
−ν+1

= HR
xx
(5)
R
yy
≡ E

y
k:k−N
f
+1
y
H
k:k−N
f
+1

= HR
xx
H
H
+ R
nn
(6)
III. SPARSE CHANNEL SHORTENING
A. Algorithm Derivation
The system block diagram is shown in Fig. 1 where our
goal is to design the length-(lN
f
) fractionally-spaced CSE,
W, and the length-(N
f
+ν) TIR, b, which minimize the mean
square of the error signal, e
k
. In other words, the CSE is
to be designed such that the overall impulse response of the
channel, h, and the CSE, W, best approximates a TIR with
few, namely (N
b
+ 1), taps. Then, the ML or MAP detectors
are designed based on the new short TIR. Although the length
of the TIR vector b is (N
f
+ ν), only (N
b
+ 1) of its taps
are designed to be nonzero. The choice of N
b
represents a
performance-complexity tradeoff. In [5], the nonzero (N
b
+1)
taps were chosen to be contiguous and their location within
the (N
f
+ν)-span of b was optimized. In this paper, we relax
the contiguousness constraint and allow the nonzero (N
b
+1)
taps to be anywhere within the (N
f
+ν)-span. Following this
approach, we show that better performance can be achieved
without increasing complexity. From Fig. 1, the error sequence
e
k
is given by [5]
e
k
= W
H
y
k:k−N
f
+1
−b
H
x
k:k−N
f
−ν+1
(7)
The MSE is formulated as
MSE ξ = E

2
2
(12)
where
¯
U is formed by all the columns of U except for the i
th
column, u
i
is the i
th
column of U, and
¯
b is formed by all the
elements of b except for the i
th
unity element. Observe that
the length-(N
f
+ ν − 1) vector
¯
b contains only N
b
nonzero
taps whose locations and values need to be determined such
that ξ is minimized. To achieve this goal, we formulate the
following convex optimization problem
min
¯
b

¯
b
1
subject to
¯
U
¯
b + u
i

2
≤
ch
(13)
3
where minimizing the l
1
-norm
¯
b
1
is equivalent to ﬁnd-
ing the sparsest solution for
¯
b as shown in CS theory [8],
[9]. Hence, the convex optimization program in (13) can be
interpreted as follows:
Find the locations and values of the taps of the sparsest TIR
such that the MSE does not exceed a given value, namely,
ch
.
Although
¯
U is not a wide matrix as it is typically the case
in CS theory formulations, the concept of minimizing the l
1
-
norm can still be used. Through (13), the design controls
the performance via
ch
. However, in this case, the designer
has no direct control on the resulting number of nonzero
taps N
b
. In situations where a speciﬁc N
b
is desired (e.g.
due to complexity constraints), the convex program in (13)
is solved via one of the matching pursuit (MP) techniques
(e.g. orthogonal MP (OMP) [17]) where the desired N
b
is
used along with
¯
U and u
i
to compute
¯
b that best matches
¯
U
¯
b
to u
i
and, hence, minimizes the MSE,
¯
U
¯
b + u
i

2
2
. After
computing
¯
b, we construct the sparse TIR, denoted by b
s
, by
simply inserting the unit tap in the i
th
location. Finally, the
optimum (in the MMSE sense) CSE taps are determined from
(9) to be
W
opt
= R
−1
yy
R
yx
b
s
, (14)
and the resulting MMSE is
ξ
min
= b
H
s
R
xx
b
s
−b
H
s
R
H
yx
R
−1
yy
R
yx
b
s
. (15)
Note that the unit tap index (i) needs to be optimized such
that the resulting MSE is minimized. However, unlike [5],
we do not need to additionally optimize the location (i.e.
starting index which is commonly known as the decision delay
parameter) of the contiguous taps because we do not constrain
the taps to be contiguous. Instead, we use the convex program
in (13) to compute their locations and values.
B. Performance Analysis
In this section, we prove that the MMSE resulting by allow-
ing the TIR taps to be noncontiguous is smaller than or equal
the MMSE when they are contiguous. Assume the indices of
the nonzero taps are known to be I = {i, i
1
, i
2
, .., i
N
b
}. We
form the following Lagrangian cost function to compute the
optimum taps values subject to the UTC
L = b
H
R
⊥
x/y
b + λ

b
H
e
i,N
f
+ν
−1

= b
H
I
R
I
b
I
+ λ

b
H
I
e
i

,N
b
+1
−1
(16)
where b
I
is the vector containing the values of the nonzero
TIR taps, i

is the unit tap index within b
I
, and R
I
is a
submatrix of R
⊥
x/y
formed by the intersections of the rows and
columns whose indices are those in I. The parameter λ is the
Lagrangian multiplier, and e
i,N
f
+ν
and e
i

,N
b
+1
are the i
th
and
i

th
columns of I
N
f
+ν
and I
N
b
+1
, respectively. Differentiating
L w.r.t. b
I
and setting the result to zero, we get
b
I,opt
=
R
−1
I
e
i

,N
b
+1
R
−1
I
(i

, i

)
(17)
where R
−1
I
(i

, i

) is the element located in the i

th
row and
the i

th
column of R
−1
I
. Substituting for b
I,opt
in (10), we get
the following MMSE
MMSE =
1
R
−1
I
(i

, i

)
(18)
We observe from (18) that the MMSE is controlled by the
indices in I. If the indices in I are not constrained to be
contiguous, then the rows and columns forming R
I
are also
not constrained to be contiguous. Due to this extra degree of
freedom, the noncontiguous solution enjoys a larger search
space and, hence, the resulting MMSE will be less than (or
at least the same as) the MMSE of the contiguous solution.
The search operation is implemented by implicitly solving the
convex optimization program in (13).
IV. SPARSE EQUALIZATION
In general, the MMSE CSE, W
opt
, in (14) is non-sparse, i.e.
all the tap weights have nonzero values. Furthermore, long
CSEs are needed to achieve a good performance especially
for highly dispersive communication channels. This increases
the complexity of computing and implementing (i.e. ﬁltering
the received signal) non-sparse FIR CSEs. Motivated by these
considerations, we propose a sparse implementation for the
FIR CSE, W, using CS theory. After computing the sparse
TIR, b
s
, we write the MSE as a function of W as follows
ξ(W) = W
H
R
yy
W−W
H
R
yx
b
s
. .. .
:=q
−b
H
s
R
H
yx
W+b
H
s
R
xx
b
s
(19)
By deﬁning the Cholesky factorization of R
yy
as R
yy
= LL
H
where L is a lower-triangular matrix, we can rewrite (19) as
ξ(W) = W
H
LL
H
W−W
H
LL
−1
q −q
H
L
−H
L
H
W + b
H
s
R
xx
b
s
(20)
where q is deﬁned in (19) and (.)
−H
≡ ((.)
H
)
−1
. Completing
the squares in (20) yields
ξ(W) = b
H
s
R
xx
b
s
−q
H
R
−1
yy
q
. .. .
=ξmin in (15)
+ L
H
W−L
−1
q
2
2
. .. .
≡ξexcess(W)
(21)
W controls ξ(W) only via the term ξ
excess
(W) because ξ
min
does
not depend on W. Since ξ
excess
(W) ≥ 0, ξ(W) is minimized by
choosing W such that ξ
excess
(W) = 0 and, hence, ξ(W) = ξ
min
.
This yields the MMSE non-sparse solution in (14) where the
implementation complexity is high. On the other hand, any
choice of W different from W
opt
makes ξ
excess
(W) > 0 which
translates into performance degradation (i.e. MSE increases).
A practical performance-complexity trade-off can be achieved
if we design the sparsest W such that ξ
excess
(W) ≤
eqz
where

eqz
controls the acceptable performance loss. According to CS
theory, this can be achieved by solving the following convex
optimization program
min
Ws
W
s

1
subject to L
H
W
s
−L
−1
q
2
2
≤
eqz
(22)
Since L is a lower-triangular matrix, the vector L
−1
q can be
easily computed using the forward substitution method [16].
Note that the optimization programs in (13) and (22) are solved
each time the CIR estimate is updated.
Assuming the input symbols are uncorrelated with the same
energy S
x
E

eqz
is computed based on the acceptable γ
max
and, then, CS
theory is applied to compute the sparsest W
s
through (22).
V. SIMULATION RESULTS AND DISCUSSION
We simulate the performance of our proposed sparse design
techniques for two CIRs. We denote the number of nonzero
CIR taps by L
ch
. The ﬁrst CIR has L
ch
= 10 contiguous taps
with all taps being Gaussian distributed with zero mean and
with the same average power. This class of channels is rather
difﬁcult to shorten because its power-delay proﬁle (PDP) is
uniform and non-sparse, so we consider this channel as a
worst case scenario and refer to it as the uniform PDP (UPDP)
CIR. The second one is the ITU Vehicular A [18] CIR which
is sparse and has L
ch
= 6 nonzero taps spanning about 13
symbols each of duration 200ns. The complex additive noise
is assumed Gaussian and white with one-sided power spectral
density level denoted by N
o
. The input SNR is deﬁned as
SNR
I
≡

Sx
No

.
Fig. 2 depicts the variations of the output SNR with the unit
tap index (i) and the decision delay for noncontiguous and
contiguous TIR designs, respectively, for the ITU Vehicular A
CIR. It is clear that the unit tap index and the decision delay
should be chosen carefully since suboptimum choices may
degrade the performance signiﬁcantly. However, we observe
that a near optimum performance is achieved as long as these
parameters are chosen in the vicinity of (N
f
+ ν)/2. In the
0 20 40 60 80 100 120
0
2
4
6
8
10
12
14
16
18
20
Unit tap index ( i )
S
N
R
o
(

Contiguous TIR design
Sparse noncontiguous TIR design
N
b
= 1 to 5
Fig. 4. SNRo(Wopt ) versus N
f
for Vehicular A CIR with SNR
I
= 20 dB.
rest of the simulations, we optimize these parameters for the
purpose of comparison between the two TIR designs.
Next, we compare the performance of our sparse channel
shortening approach with the contiguous approach in [5]
using the MMSE non-sparse CSE. In Figs. 3 and 4, we plot
SNR
o
(W
opt
) versus N
f
for the UPDP CIR and the Vehicular
A CIR, respectively, and for N
b
ranging from 1 (lower curve)
to L
ch
− 1 (upper curve). Recall that (N
b
+ 1) represents the
number of taps in the TIR b. As N
b
increases, the output SNR
increases for both sparse and contiguous designs of the TIR as
expected which means that the TIR becomes more accurate in
approximating the actual channel. However, we observe that
our sparse noncontiguous design of the TIR performs better
than the conventional contiguous MMSE design for all choices
of N
f
and N
b
. In Fig. 3, the sparse noncontiguous design
with N
b
= 3 achieves a higher output SNR than that achieved
by the contiguous MMSE design with N
b
= 5 almost over
the whole range of N
f
. Therefore, allowing the TIR taps to
be noncontiguous results in a better performance at the same
complexity or in a lower complexity at the same performance.
Next, we study the effect of sparse CSE design on per-
formance where we plot the active CSE taps (i.e those with
nonzero weights) percentage of the total equalizer span N
f
versus γ
max
which is the maximum loss in the output SNR.
This study is conducted in Figs. 5 and 6 for the UPDP CIR
and the Vehicular A CIR, respectively, with N
f
= 100 and N
b
5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
10
20
30
40
50
60
70
80
90
100
γ
max
(dB)
P
e
r
c
e
n
t
a
g
e

o
f

C
S
E

a
c
t
i
v
e

t
a
p
s

(
%
)

Sparse noncontiguous TIR design
Contiguous TIR design
N
b
= 1 to 9
Fig. 5. Active CSE taps percentage versus γmax for UPDP CIR with SNR
I
=
20 dB and total CSE span of N
f
= 100.
ranging from 1 (upper curve) to L
ch
−1 (lower curve). Again,
contiguous and sparse designs of the TIR are compared. Four
observations are made based on Figs. 5 and 6. First, allowing
a higher loss in the output SNR yields a bigger reduction
in the number of CSE taps. Second, the active CSE taps
percentage increases as N
b
decreases because the equalizer
has to work harder (i.e. needs more taps) to shorten the CIR to
a shorter TIR. Third, our sparse noncontiguous design results
is bigger reduction in the number of CSE taps compared to
the contiguous design at the same output SNR loss because
the former design better approximates the actual channel, so
the CSE needs fewer taps to accomplish its mission. Fourth,
Comparing Figs. 5 and 6, we ﬁnd that the amount of reduction
in the number of CSE taps is bigger for the Vehicular A CIR
than for the UPDP CIR. This is because the Vehicular A CIR
is sparse and has fewer nonzero taps than the UPDP CIR, so
the CSE can shorten its CIR using fewer taps. This ensures
that the UPDP CIR represents a worst case scenario. In Fig. 5,
we observe that allowing a maximum of 0.4 dB loss in SNR
o
for N
b
= 3 results in about 64% reduction in the number of
CSE taps, i.e. the equalizer can shorten the channel using only
36 taps out of a total span of N
f
= 100 taps. Inspecting the
bold curve representing N
b
= 3 in Fig. 3, we ﬁnd that the
achieved SNR
o
, even after this 0.4 dB loss, is still about 1
dB higher than the achieved SNR
o
by the MMSE non-sparse
equalizer with N
f
= 36. The proposed design superiority in
terms of bit error rate is shown in [19].
VI. CONCLUSION
We proposed a novel approach based on CS theory for
sparse FIR channel shortening design. The key idea behind the
new approach is to allow the target channel taps to be noncon-
tiguous unlike previous work where they were constrained to
be contiguous. The new noncontiguous TIR design was proved
analytically and illustrated via simulations to yield a lower
MSE compared to the contiguous design. Furthermore, we
proposed a new l
1
-norm-based convex formulation to design
sparse FIR channel shortening equalizers. Simulation results
demonstrated substantial complexity reductions at a small
performance loss.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
0
10
20
30
40
50
60
70
80
90
100
γ
max
(dB)
P
e
r
c
e
n
t
a
g
e